Abstract
In this article, the authors apply the supersolution and subsolution methods, instead of energy estimate methods, to investigate the critical extinction exponents for a fast diffusion equation with a nonlocal source and an absorption term. They give a classification of the exponents and coefficients for the solutions to vanish in finite time or not, which improve, in some sense, the results by Xu et al. (Bound. Value Probl. 2013:24 2013) and by Han et al. (Arch. Math. 97:353363, 2011).
MSC: 35K55, 35K57.
Keywords:
fast diffusion equation; critical exponents; extinction; supersolutions and subsolutions1 Introduction
In this paper, we investigate the following fast diffusion equation with a nonlocal source and an absorption term:
where , , Ω is a bounded domain in () with smooth boundary ∂Ω, and is a nonnegative nontrivial function.
The equation in (1.1) is a fast diffusion equation perturbed by both a nonlocal source term and an absorption term, which describes the diffusion of concentration of some Newtonian fluids or the density of some biological species (see [1,2] and the references therein). What we are interested in here is the extinction in finite time of the nonnegative solutions of (1.1), i.e. there exists a finite time such that the solution is nontrivial for , but for almost every . In this case, T is called the extinction time. As one of the most important properties of solutions of evolutionary equations, extinction in finite time of solutions has been intensively studied by several authors (see [316] and the references therein). In particular, in a recent paper by Xu et al.[13], the authors investigated the extinction and nonextinction phenomena of solutions of Problem (1.1) and obtained the following result.
Theorem 1.1 (Theorems 13 in [13])
Suppose. Ifand, then the nonnegative nontrivial weak solution of Problem (1.1) vanishes in finite time for any nonnegative initial data provided that eitherorais sufficiently small; If, then the nonnegative nontrivial weak solution of Problem (1.1) vanishes in finite time provided that, orais sufficiently small, andwithCbeing a positive constant depending only onN, randm; Ifand, then the nonnegative nontrivial weak solution of Problem (1.1) vanishes in finite time for any nonnegative initial data provided thatbis sufficiently large.
It can be seen from the above theorem that the extinction of nonnegative nontrivial weak solutions to Problem (1.1) occurs when the absorption term is in some sense strong. However, when the absorption term is suitably weak, whether Problem (1.1) admits nonextinction solutions or not is not answered in [13]. On the other hand, it can be seen from [9] that is the critical extinction exponent when there is no absorption term. An interesting problem is whether the absorption term can change the critical extinction exponent. We know from a recent paper [17] by Liu et al. that the critical extinction exponent is not changed (at least when the source is local) when . However, when the absorption term is nonlinear, i.e. when , the problem is open.
Motivated by the works mentioned above, we investigate the critical extinction exponents for Problem (1.1) by constructing suitable super and subsolutions and give a more complete classification of exponents and coefficients for the solutions to vanish in finite time or not.
Let be the unique positive solution of the following linear elliptic problem:
Throughout this paper, we denote
By the strong maximum principle we know that . Our main results are the following theorems.
Theorem 1.2If, then all the solutionsvanish in finite time for suitably small initial data. Ifand, then all the solutionsvanish in finite time for any nonnegative bounded initial data.
Theorem 1.3Iforwith, then all the solutionsof Problem (1.1) vanish in finite time for appropriately small initial data.
Theorem 1.4Iforwith, then Problem (1.1) admits at least one nonextinction solution for any nonnegative initial data.
Here will be given in the proof of Theorem 1.4.
Theorem 1.5Ifwith, then Problem (1.1) admits at least one nonextinction solution for any nonnegative initial data.
Theorem 1.6Ifwith, then (1.1) admits at least one nonextinction solutionfor any strictly positive initial data. Ifwith, then Problem (1.1) admits at least one extinction solutionfor any nonnegative initial data. Ifand, thenvanishes in the sense that.
Remark 1.1 Comparing Theorems 1.21.6 with Theorem 1.1 we can see that our results complement those obtained in [13] since the case is also considered in our paper. Moreover, according to Theorems 1.2, 1.5 and the first part of Theorem 1.6 it is easy to see that is the critical extinction exponent for Problem (1.1) when , which is the same as the problems with local reaction terms (see [17]). However, when and , for the nonlocal problem under consideration, the first eigenvalue of −Δ in Ω no longer plays the same role as it does in the local case.
2 Proof of the main results
It is well known that the equation in (1.1) is degenerate if and singular if , and therefore there is no classical solution in general. To state the definition of the weak solution, we first define the class of nonnegative testing functions
Definition 2.1 A function is called a subsolution (supersolution) of Problem (1.1) in if the following conditions hold:
(iii) for almost every and every ,
A function is called a local solution of (1.1) if it is both a subsolution and a supersolution for some and is called a solution of (1.1) if it is a local solution of (1.1) in for any .
Local existence of weak solutions of (1.1) can be obtained by utilizing the methods of standard regularization (see [9]) and the continuity of the solutions can be derived by the arguments similar to that in [18]. Moreover, Problem (1.1) admits global solutions when the initial data are small (see [1]). Since the regularization procedure is crucial in what follows, we shall sketch the outline. Consider the regularized problem
where may be chosen sufficiently small in such a way that there exists a solution of (2.2) on for every , and is bounded independently of k. Furthermore, for , and a supersolution (subsolution) comparison theory holds for (2.2) (see [1,19]).
Since is monotone in k, we may define , and it is easy to see that is a solution of (1.1). Furthermore, if u is a solution of (1.1), then we have
where we use the fact on ∂Ω to derive this inequality. With and defined so that
and
we have
Thus, we can choose the appropriate test function ξ as in [1,19] to obtain . If u is a subsolution of (1.1), the above argument shows that . Thus is the maximal solution of (1.1), and this solution satisfies a subsolution comparison principle.
Before proving our main results, we give a comparison principle for the solution of Problem (1.1), which is similar to Proposition 2.3 in [9] and can be proved by modifying the above arguments (see also [1,10,19]).
Proposition 2.1Letuandvbe a nonnegative bounded subsolution and a nonnegative supersolution of (1.1), respectively. If eitheranduis bounded from the above orandvhas a positive lower bound, theninifin Ω.
Proof of Theorem 1.2 Case (i): with . For any bounded smooth domain such that , let be the unique solution of the following elliptic problem:
By the comparison principle for linear elliptic problem we know in Ω. Set , and . It is well known from the strong maximum principle that .
By continuity, we can choose a suitable domain with such that . Define , where satisfies
Since , it follows from the theory in ODEs that is nonincreasing and for all
Then it can be verified that is a supersolution of (1.1). In fact, because and , we know that satisfies the following inequalities (in the weak sense):
In addition, on , for any , and by the choice of A. Moreover, there exists a positive constant such that in . Therefore, by applying Proposition 2.1 to (1.1) we see that for , which implies . The arbitrariness of and ensure that . Furthermore, let , then satisfies (1.1) with the initial condition . By the aforementioned proof, we see that with any . From the relation of the extinction time of to A, it follows that for any , i.e. for any .
Case (ii): . Let ϕ, be the same as Case (i) and denote . Set with , then it is easy to verify that is a supersolution of (1.1) when is sufficiently small such that in Ω. Applying Proposition 2.1 to Problem (1.1) in for any we obtain in , which implies that . Therefore, satisfies
By the choice of k and it is easily verified that . Thus, by the results of Case (i), we can conclude that the solution vanishes in finite time when the initial data are suitably small. The proof of this theorem is complete. □
Proof of Theorem 1.3 We first prove the case with . Set where satisfies the following ordinary differential equation:
Since and , we know by integrating the ODE that vanishes at some finite time . Moreover, as in the proof of Theorem 1.2, it can be verified that is a supersolution of (1.1). Thus, by applying Proposition 2.1 to and for any we know that also vanishes at .
In the case , let satisfy the following ODE:
where . Similar to the first case, it is well known that vanishes in finite time since and is a supersolution of (1.1) provided that is small enough such that . Applying Proposition 2.1 to and guarantees the finite time extinction of . This completes the proof of Theorem 1.3. □
Proof of Theorem 1.4 (i) Consider first the case . Let be the first eigenvalue of the following eigenvalue problem:
and () be the corresponding eigenfunction. We may normalize such that . Denote and let satisfy the ODE problem
It is easy to check that is nondecreasing and bounded from above by . Set . We shall show that is a subsolution of (1.1) when is sufficiently small. In fact, simple computations show that
and
For to be a subsolution of (1.1), it suffices to show that
which follows from
where . It is easy to see that (2.10) is valid for sufficiently small since .
Next, we turn our attention to construct a supersolution of (1.1). Set , where . Then it is not hard to see that is a supersolution and . Therefore, by an iteration process, one can obtain a solution of Problem (1.1), which satisfies . Indeed, define and iteratively to be a solution of the problem
subject to the boundary and initial conditions as that in (1.1). By applying the comparison technique used in the proof of Lemma 2.1 in [1,12] we know that the function , for every and , is a solution of (1.1). Because does not vanish, neither does .
(ii) The case can be treated similarly to Case (i).
(iii) Finally we consider the case with . Let satisfy the following ODE:
Then is nondecreasing and satisfies . (The upper bound of can be obtained by contradiction arguments and the monotonicity of follows immediately as the upper bound is derived.) As in the proof of Case (i), we can construct a nonextinction subsolution with sufficiently small.
To construct a supersolution, consider the following eigenvalue problem:
where is a bounded domain with smooth boundary . Let and () be its first eigenvalue and the corresponding eigenfunction, respectively. We may normalize such that . Denote and . Set , then we shall show that is a supersolution of (1.1) provided that is suitably large. Indeed, if , we know that on , in Ω and satisfies the following inequalities (in the weak sense):
Moreover, by the choice of k. Therefore, by applying the monotonicity iteration process we can obtain a nonextinction solution of (1.1) satisfying . The proof of Theorem 1.4 is complete. □
Proof of Theorem 1.5 The proof of this theorem is similar to that of Theorem 1.4, so we only sketch the outline here. Set where is defined in (1.2) and satisfies the following ODE problem:
Since and , it is well known that is nondecreasing and bounded above by . Then is a subsolution of (1.1) if is sufficiently small. On the other hand, the supersolution can be chosen to be a large positive constant L satisfying . It can be observed that is a pair of subsolution and supersolution of (1.1) satisfying . Therefore, by monotonicity iteration, we know that (1.1) admits at least one solution such that . Since in , cannot vanish at any finite time. The proof of Theorem 1.5 is complete. □
Proof of Theorem 1.6 (i) Let be any solution of (1.1). It can be verified that, for the case , a sufficiently large constant L is a supersolution of (1.1). Therefore, we know that in . For convenience, in the following proof, we assume that the weak solution is appropriately smooth; otherwise, we can consider the corresponding regularized problem, and the same result can also be obtained through an approximate process (see [15]). Multiplying equation (1.1) by and integrating by parts over Ω yield the identity
Recall the embedding theorem
Combining this result with (2.13) and using Hölder’s inequality on the right hand side of (2.13) one obtains
Noticing that and , we see from (2.14) that
which implies
This shows that tends to 0 exponentially as .
Since , is nonincreasing and for . Noticing and , one can see that is a supersolution of (1.1) provided that in Ω. By using the arguments similar to that of the proof of Case (i) of Theorem 1.2 we can show that any solution of Problem (1.1) vanishes in finite time.
(iii) Finally we consider the case . First we construct a nonextinction subsolution of (1.1). Set , where , α are two positive constants to be determined. Noticing that , it is easily verified that when , is a subsolution of (1.1) if and if is so small such that . When , for to be subsolution of (1.1) it is reasonable to choose first so small such that and then . Next, since and is bounded, we can choose a sufficiently large constant to be a supsolution of (1.1). Therefore, by monotonicity iteration, we can obtain a solution of (1.1) satisfying . Since does not vanish at any finite time, neither does . The proof of Theorem 1.6 is complete. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions which improve the original manuscript. The authors would also like to express their sincere gratitude to Professor Wenjie Gao for his enthusiastic guidance and constant encouragement. The project was supported by NSFC (11271154).
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